on some classes of discrete polynomials and ordinary ... · on some classes of discrete polynomials...

arX

iv:1

308.

5018

v3 [

nlin

.SI]

24

Mar

201

4

On some classes of discrete polynomials and

ordinary difference equations

Andrei K Svinin

Institute for System Dynamics and Control Theory, Siberian Branch of Russian

Academy of Sciences, Russia

E-mail: [email protected]

Abstract. We introduce two classes of discrete polynomials and use them for

constructing ordinary difference equations admitting a Lax representation in terms

of these polynomials. We also construct lattice integrable hierarchies in their explicit

form and show some examples.

PACS numbers: 02.30.Ik

Keywords: KP hierarchy, integrable lattices

Submitted to: J. Phys. A: Math. Theor.

1. Introduction

A discrete counterpart of an Nth order autonomous ordinary differential equation is a

recurrence relation of the form

T (i+N) = F (T (i), T (i+ 1), . . . , T (i+N − 1)) (1)

for an unknown function T (i) of a discrete variable i ∈ Z. One calls (1) an ordinary

discrete (difference) equation of Nth order. Equation (1) yields a map RN → R

N for

real-valued initial data {yj ≡ T (i0 + j) : j = 1, . . . , N − 1}. By analogy with ordinary

differential equations, the function J = J(i) = J(T (i), T (i + 1), . . . , T (i + N − 1))

is called the first integral for difference equation (1) if by virtue of this equation one

has J(i + 1) = J(i). There are ordinary difference equations which have some special

properties which yield a regular behavior of their solutions. One of such properties is

complete or Liouville-Arnold integrability [8], [32], that is, the existence of a sufficient

number of functionally independent integrals which are in involution with respect to

a Poisson bracket. Usually such integrable equations are grouped into some infinite

classes of their own kind. Examples to mention are the sine-Gordon, modified KdV,

potential KdV and Lyness equations [17], [18], [29], [30]. It is worth remarking that these

equations can be obtained as special reductions of partial difference equations for an

http://arxiv.org/abs/1308.5018v3

On some classes of discrete polynomials and ordinary difference equations 2

unknown function of two discrete variables. Studies carried out, for example, in [15], [31]

shows that proving the Liouville-Arnold integrability for the map under consideration

is a quite difficult and complicated task. Another characteristic of discrete equations

claiming to have an adjective ‘integrable’ is a Lax pair representation. This means that

the system of equations appear as a condition of compatibility of two linear equations

with some spectral parameters. Let us remark that the above mentioned difference

equations in fact possess this property. It should be related to the complete integrability,

how most likely it can be considered just as an indicator of Liouville-Arnold integrability.

Also possible integrability criteria which might be applied to discrete equations are the

zero algebraic entropy [2], [14], [33] and singularity confinement [13].

The main purpose of our paper is to exhibit some classes of ordinary autonomous

difference equations having a Lax pair representation. In the general case, we obtain

multi-field difference systems which might be completely integrable. Also we present

an approach which allows us to construct many integrable hierarchies of evolution

differential-difference equations in their explicit form. This goal is achieved by solving

recurrence equations on the coefficients of formal pseudo-difference operators in terms

of which we write the Lax pairs. As a result we obtain some polynomials T ks and

Sks which, in a sense, generalize elementary and complete symmetric polynomials [20],

respectively. Therefore, we construct our ordinary difference equations and hierarchies

of evolution differential-difference equations in terms of these polynomials. Then we

compare these studies with our approach given in [23], [24], [25], [26] which relates many

lattice integrable hierarchies to the Kadomtsev-Petviashvili (KP) hierarchy and the nth

discrete KP hierarchy. It is worth remarking that this approach goes back to [19]. In

this framework lattice integrable hierarchies appear as reductions of the nth discrete KP

hierarchy on corresponding invariant submanifolds. It allows us to show the meaning

of obtained ordinary difference equations from this point of view. Namely, they turn

out to be algebraic constraints compatible with corresponding integrable hierarchies

which yield some invariant submanifolds in the solution space of the system under

consideration. Looking ahead let us show the simplest example — the Volterra lattice

hierarchy of evolution equations‡

∂sT (i) = (−1)sT (i) {Sss(i− s+ 2)− Ss

s(i− s)} , s ≥ 1,

whose right-hand sides are defined by the following discrete polynomials

S11(i) = T (i), S2

2(i) = T (i+ 1) {T (i) + T (i+ 1) + T (i+ 2)} ,

S33(i) = T (i+ 2) {T (i+ 1) {T (i) + T (i+ 1) + T (i+ 2) + T (i+ 3)}

+T (i+ 2) {T (i+ 1) + T (i+ 2) + T (i+ 3)}

+T (i+ 3) {T (i+ 2) + T (i+ 3)}+ T (i+ 4)T (i+ 3)}

‡ Here the symbol ∂s stands for derivative ∂/∂ts with respect to the evolution parameter ts.


and so on. This hierarchy corresponds to the invariant submanifoldM1,2,1 of the discrete

KP hierarchy [27]. Our approach gives an infinite number of constraints

Sks+1(i)T (i+ k) = Sk

s+1(i+ 1)T (i+ s), s ≥ k + 1, k ≥ 0

each of which represents an autonomous difference equation (1) of the order N = k + s

with the right-hand side rationally depending on its arguments and some number of

parameters. In particular, in the case k = 0, it is specified as the periodicity condition

T (i+ s) = T (i).

The rest of the paper is organized as follows. In section 2, in the framework of

formal pseudo-difference operators, we construct two classes of above-mentioned quasi-

homogeneous discrete polynomials of an infinite number of the fields {T j = T j(i) : j ≥

1}. All statements in this section we formulate in the language of corresponding multi-

variate polynomials. In section 3, we apply these polynomials for constructing ordinary

difference equations which arise in this approach as a conditions of the consistency of

two linear equations yielding the Lax pairs. We also construct integrable hierarchies of

evolution differential-difference equations in their explicit form and provide the reader

by well-known examples. In section 4, we set out our approach to integrable lattices

from the point of view of the KP hierarchy and derive the same classes of ordinary

difference equations in this framework. This enable us to explain the meaning of the

difference equations obtained in section 3 as some algebraic constraints compatible with

the flows of corresponding integrable hierarchies.

2. Discrete polynomials T ks and Sk

s

The main goal of this section is to describe some quasi-homogeneous polynomials

associated with formal pseudo-difference operators. This will enable us to construct

ordinary difference equations admitting a Lax representation and integrable hierarchies

of lattice evolution equations in their explicit form.

2.1. Formal pseudo-difference operators and polynomials T ks and Sk

s

Given an infinite number of unknown functions {T j = T j(i) : j ≥ 1} of a discrete

variable i ∈ Z and an arbitrary pair of co-prime integers h ≥ 1 and n ≥ 1, we consider

a pseudo-difference operator§

T1 = Λ−h +∑

j≥1

z−jT j(i− (j − 1)n− h)Λ−h−jn

and its powers‖

Ts ≡ (T1)s = Λ−sh +

∑

j≥1

z−jT js (i− (j − 1)n− sh)Λ−sh−jn (2)

§ Here Λ stands for the shift operator acting as Λ(f(i)) = f(i + 1).‖ Strictly speaking, we have to indicate dependence of polynomials T k

s and Sks on n and h writing, for

example T(n,h,k)s and S

(n,h,k)s but we believe that such notations are quite cumbersome and prefer to

use simplified ones in the hope that this does not lead to a confusion.


for all s ≥ 1. By definition, the coefficients T ks , for all s ≥ 1 are some discrete

polynomials of infinite number of fields T j. We indicate this as T ks = T k

s [T1, . . . , T k].¶

By definition, Ts1+s2 = Ts1Ts2 = Ts2Ts1. In terms of the coefficients of pseudo-

difference operators (2) this looks as a pair of two compatible identities

T ks1+s2

(i) = T ks1(i+ s2h) +

k−1∑

j=1

T js2(i)T k−j

s1(i+ s2h+ jn) + T k

s2(i)

= T ks2(i+ s1h) +

k−1∑

j=1

T js1(i)T k−j

s2(i+ s1h+ jn) + T k

s1(i).

In particular, one sees that these polynomials must satisfy a compatible pair of

recurrence relations

T ks (i) = T k

s−1(i+ h) +

k−1∑

j=1

T j(i)T k−js−1 (i+ h + jn) + T k(i) (3)

= T ks−1(i) +

k−1∑

j=1

T j(i+ (s− 1)h+ (k − j)n)T k−js−1 (i) + T k(i+ (s− 1)h). (4)

Taking for use one of them, one derives all polynomials T ks starting from T k

1 = T k.

Some remarks are in order. In practice, it is more convenient to work with multi-

variate polynomials Q{yrj} rather than with discrete ones identifying yrj = T r(i+ j) for

r ≥ 1 and j ∈ Z. In fact, it is not only the additional notation. We believe that multi-

variate polynomials which we present below could be more fundamental objects than

their discrete counterparts which are used throughout the paper for constructing discrete

integrable systems having Lax pair representation. Therefore, let us formulate in this

section all propositions concerning discrete polynomials in the language of multi-variate

polynomials.

Let k = [ykj ] be a scaling dimension or weight. For monomials one putsr∑

j=1

kj = [yk1j1 yk2j2. . . ykrjr ].

One says that a polynomial Q = Q{yrj} is quasi-homogeneous one of the degree k if all

monomials entering in this polynomial have the same weight k or in other words that

Q{λryrj} = λkQ{yrj} for any λ ∈ R.

Let us rewrite relations (3) and (4) as equivalent relations for multi-variate

polynomials T ks {y

rj}

+

T ks = T k,h

s−1 +

k−1∑

j=1

yj0Tk−j,h+jns−1 + yk0 (5)

= T ks−1 +

k−1∑

j=1

yj(s−1)h+(k−j)nTk−js−1 + yk(s−1)h. (6)

¶ We use the term ‘discrete polynomial’ by analogy with the notion of differential polynomial.+ The notations like T k,α

s and Sk,αs stand for α-shifted polynomials, that is, T k,α

s ≡ T ks {y

rj+α} and

Sk,αs ≡ Sk

s {yrj+α}, respectively.


We can solve (5) and (6) to obtain the following.

Proposition 1 A solution of (5) and (6) with initial condition T k1 = yk0 for k ≥ 1 is

given by an infinite number of quasi-homogeneous polynomials

T ks =

∑

K

∑

0≤λ1<···<λp≤s−1

yk1λ1hyk2λ2h+k1n

· · · ykp

λph+(k1+···+kp−1)n. (7)

Let us remark that the summation in (7) is performed over all compositions K =

(k1, . . . , kp) of number k with kj ≥ 1.

Proof of proposition 1. Firstly, we remark that (7) gives T k1 = yk0 for all k ≥ 1.

Next, we observe that the polynomials T ks = T k

s {yrj} are uniquely defined from these

recurrence relations starting from T k1 = yk0 . Thus, to prove this proposition we have to

show that the polynomials T ks defined by explicit expression (7), solve (5) and (6). To

this aim, we consider the partition of the set Dp,s ≡ {λj : 0 ≤ λ1 < · · · < λp ≤ s − 1}

into two non-intersecting subsets D(1)p,s ≡ {λj : λ1 = 0; 1 ≤ λ2 < · · · < λp ≤ s − 1} and

D(2)p,s ≡ {λj : 1 ≤ λ1 < · · · < λp ≤ s− 1}. It is evident that

T k,hs−1 =

∑

K

∑

{λj}∈D(2)p,s

yk1λ1hyk2λ2h+k1n

· · · ykp

λph+(k1+···+kp−1)n.

Moreover∑

K

∑

{λj}∈D(1)p,s

yk1λ1hyk2λ2h+k1n

· · · ykp

λph+(k1+···+kp−1)n

=∑

K

yk10∑

1≤λ2<···<λp≤s−1

yk2λ2h+k1n· · · y

kp

λph+(k1+···+kp−1)n.

It is obvious that the set K of all compositions of number k can be presented as

K =⊔k

j=1Kj , where Kj stands for the set of compositions of k with k1 = j. Clearly, if

K ∈ Kj then the set (k2, . . . , kp) presents a composition of number k − j. Taking this

into account, we obtain

∑

K

yk10∑

1≤λ2<···<λp≤s−1


kp

λph+(k1+···+kp−1)n

=k∑

j=1

yj0∑

(k2,...,kp)

∑

1≤λ2<···<λp≤s−1


kp

λph+(k1+···+kp−1)n

=

k∑

j=1

yj0Tk−j,h+jns−1 .

Therefore we proved that (5) is an identity for polynomials defined by (7). To prove

that (6) is also an identity for (7), we use similar reasonings. Therefore, the proposition

is proved. �

Consider now a pseudo-difference operator

S1 ≡ (T1)−1 = Λh +

∑

j≥1

(−1)jz−jSj(i− (j − 1)n)Λh−jn


and its powers

Ss ≡ (S1)s = Λsh +

∑

j≥1

(−1)jz−jSjs(i− (j − 1)n)Λsh−jn.

Clearly, each coefficient Sks is some discrete polynomial of the fields {T 1, . . . , T k}. Using

obvious relations Ss1−s2 = Ss1Ts2 = Ts2Ss1 for s1 ≥ s2 we get

Sks1−s2

(i) = Sks1(i) +

k−1∑

j=1

(−1)jT js2(i+ (s1 − s2)h)S

k−js1

(i+ jn)

+ (−1)kT ks2(i+ (s1 − s2)h) (8)

= Sks1(i− s2h) +

k−1∑

j=1

(−1)jT js2(i− s2h+ (k − j)n)Sk−j

s1(i− s2h)

+ (−1)kT ks2(i− s2h) (9)

and in particular

Sks−1(i) = Sk

s (i) +

k−1∑

j=1

(−1)jT j(i+ (s− 1)h)Sk−js (i+ jn) + (−1)kT k(i+ (s− 1)h) (10)

= Sks (i− h) +

k−1∑

j=1

(−1)jT j(i− h+ (k − j)n)Sk−js (i− h) + (−1)kT k(i− h). (11)

Let us rewrite (10) and (11) in equivalent form as relations for multi-variate polynomials

Sks {y

rj}, that is,

Sks−1 = Sk

s +

k−1∑

j=1

(−1)jyj(s−1)hSk−j,jns + (−1)kyk(s−1)h (12)

and

Sk,hs−1 = Sk

s +

k−1∑

j=1

(−1)jyj(k−j)nSk−js + (−1)kyk0 . (13)

Proposition 2 A quasi-homogeneous solution of (12) and (13) is given by

Sks = (−1)k

∑

K

(−1)p∑

0≤λ1≤···≤λp≤s−1

yk1λ1h+(k2+···+kp)n

· · · ykp−1

λp−1h+kpnykpλph. (14)

Proof. To prove that (14) satisfy (13) we consider a partition of the set Bp,s ≡ {λj :

0 ≤ λ1 ≤ · · · ≤ λp ≤ s − 1} into two non-intersecting subsets B(1)p,s ≡ {λj : λ1 = 0; 0 ≤

λ2 ≤ · · · ≤ λp ≤ s− 1} and B(2)p,s ≡ {λj : 1 ≤ λ1 ≤ · · · ≤ λp ≤ s− 1} what gives

Sks = (−1)k

∑

K

∑

{λj}∈B(2)p,s

(−1)pyk1λ1h+(k2+···+kp)n

· · · ykp−1

λp−1h+kpnykpλph

+ (−1)k∑

K

(−1)pyk1(k2+···+kp)n

∑

0≤λ2≤···≤λp≤s−1

yk2λ2h+(k3+···+kp)n

· · · ykp−1

λp−1h+kpnykpλph.


The first member in this sum gives Sk,hs−1, while using the partition K =

⊔k

j=1Kj for the

second member yields

k−1∑

j=1

(−1)j+1yj(k−j)nSk−js + (−1)k+1yk0 .

To prove that (12) is also an identity for (14), we use similar arguments. Therefore the

proposition is proved. �

Let us remark that in particular case s = 1, (14) becomes

Sk1 = (−1)k

∑

K

(−1)pyk1(k2+···+kp)n· · · y

kp−1

kpnykp0 (15)

Let us denote xrj = Sr,j1 . From (12) and (13) we obtain that

xkj +

k−1∑

q=1

(−1)qyqj+(k−q)nx

k−qj + (−1)kykj = 0 (16)

and

xkj +

k−1∑

q=1

(−1)qyjjxk−jj+qn + (−1)kykj = 0. (17)

It is evident that (17) can be obtained from (16) and vice versa by replacing yrj ↔ xrj .

This means that (15) gives an invertible involutive transformation which allows one to

express yrj as a polynomial in xrj as

ykj = (−1)k∑

K

(−1)pxk1j+(k2+···+kp)n

· · ·xkp−1

j+kpnxkpj

= (−1)k∑

K

(−1)pxk1j xk2j+k1n

· · ·xkp

j+(k1+···+kp−1)n.

Making use of the latter we can express any multi-variate polynomials Q{yrj} in variables

xrj . In particular, the following is valid.

Proposition 3 Polynomials T ks and Sk

s are expressed via the set of variables {xrj} as

Sks =

∑

K

∑

{λj}∈Dp,s

xk1λ1h+(k2+···+kp)n

· · ·xkp−1

λp−1h+kpnxkpλph

(18)

and

T ks = (−1)k

∑

K

∑

{λj}∈Bp,s

(−1)pxk1λ1hxk2λ2h+k1n

· · ·xkp

λph+(k1+···+kp−1)n. (19)

We do not give the proof of this proposition because it is very similar to that of

propositions 1 and 2. It is enough to say that (18) and (19) solve the recurrence relations

Sks = Sk

s−1 +

k−1∑

j=1

xj(s−1)hSk−j,jns−1 + xk(s−1)h (20)

= Sk,hs−1 +

k−1∑

j=1

xj(k−j)nSk−j,hs−1 + xk0 (21)


and

T ks−1 = T k

s +k−1∑

j=1

(−1)jxj(s−1)h+(k−j)nTk−js + (−1)kxk(s−1)h (22)

and

T k,hs−1 = T k

s +k−1∑

j=1

(−1)jxj0Tk−j,jns + (−1)kxk0, (23)

respectively. Thus, one sees that the variables yrj and xrj are on equal footing.

Putting s1 = s2 ≡ s with arbitrary s ≥ 1 in (8) and (9) gives the relations which

we rewrite as

Sks +

k−1∑

j=1

(−1)jT jsS

k−j,jns + (−1)kT k

s = 0 (24)

and

Sks +

k−1∑

j=1

(−1)jT j,(k−j)ns Sk−j

s + (−1)kT ks = 0. (25)

Solving these equations yields the following.

Proposition 4 Polynomials T ks and Sk

s are expressed via each other as

Sks = (−1)k

∑

K

(−1)pT k1s T k2,k1n

s · · ·T kp,(k1+···+kp−1)ns (26)

and

T ks = (−1)k

∑

K

(−1)pSk1s S

k2,k1ns · · ·Skp,(k1+···+kp−1)n

s . (27)

Proof. The proof of this proposition, in fact, follows the lines proposed in [20] to express

the elementary and complete symmetric polynomials via each other in the determinant

form with minor modification. We only remark that one can rewrite (24) and (25) as

two matrix equations SsTs = I and TsSs = I, respectively, with semi-infinite matrices

Ts =

1 0 0 0 0 . . .

T 1s 1 0 0 0 . . .

T 2s T 1,n

s 1 0 0 . . .

T 3s T 2,n

s T 1,2ns 1 0 . . .

......

......

......

and

Ss =

1 0 0 0 0 . . .

−S1s 1 0 0 0 . . .

S2s −S1,n

s 1 0 0 . . .

−S3s S2,n

s −S1,2ns 1 0 . . .

......

......

......


and I being semi-infinite identity matrix and use them the known expression for the

elements of an inverse matrix via the elements of the original one. As a result, in

particular, we obtain formulas (26) and (27), which can be presented in the determinant

form. �

2.2. Polynomials T js and Sj

s

Given an infinite number of constants cj, define

Ts ≡ Ts +∑

j≥1

z−jhcjTs+jn.

Observe that

Ts = Λ−sh +∑

j≥1

z−jT js (i− (j − 1)n− sh)Λ−sh−jn

with coefficients T js which can be described as follows. Let k = κh + r, where r stands

for the remainder of division of k by h, then

T ks (i) = T k

s (i) +κ∑

j=1

cjTk−jhs+jn (i).

Given an infinite number of constants Hj, define

Ss ≡ Ss +∑

j≥1

z−jhHjSs−jn.

Here it is supposed that s− jn ≤ 1. Observe that

Ss = Λsh +∑

j≥1

(−1)jz−jSjs(i− (j − 1)n)Λsh−jn

with

Sks (i) = Sk

s (i) +κ∑

j=1

(−1)jhHjSk−jhs−jn (i+ jhn).

It is evident that

Ts1+s2 = Ts1Ts2 = Ts2 Ts1, Ss1+s2 = Ss1Ss2 = Ss2Ss1 ,

Ts1−s2 = Ts1Ss2 = Ss2 Ts1 , Ss1−s2 = Ss1Ts2 = Ts2Ss1 .

This means, in particular, that polynomials T ks and Sk

s as well as polynomials T ks and

Sks also satisfy identities (3), (4), (10), (11), (20), (21), (22) and (23). The difference

between these solutions is that in the case of quasi-homogeneous polynomials T ks and

Sks one starts from T k

1 = T k and Sk1 given by (15), while for deriving polynomials T k

s

and Sks one uses as the initial data in the corresponding recurrence relations some linear

combinations of quasi-homogeneous polynomials.


2.3. l-reduced polynomials

Requiring yrj ≡ 0, for r ≥ l + 1, we obtain l-reduced polynomials T ks and Sk

s . For

example, 1-reduced polynomials look as

T ks =

∑

{λj}∈Dk,s

yλ1hyλ2h+n · · · yλkh+(k−1)n (28)

and

Sks =

∑

{λj}∈Bk,s

yλ1h+(k−1)n · · · yλk−1h+nyλkh (29)

where yj ≡ y1j . These polynomials were reported in [27]. One can also consider l-

reduction of polynomials (18) and (19) defined by the condition xrj ≡ 0, for r ≥ l + 1.

One sees that in the degenerate case n = 0 and h = 1 (28) and (29) are nothing but

elementary and complete symmetric polynomials, respectively [20]. We could expect

that polynomials given by (7) and (14) keep some nice properties of symmetric ones.

In this connection, the following question could be addressed: whether polynomials T ks

and Sks in the general case generate some algebraic structure?

2.4. Bibliographical remarks

It is worth mentioning that 1-reduced discrete polynomials T ks given by (28) in the

particular case h = 1 and n = 1 first appeared in [29], where they were used for

description of the integrals of the mKdV and sine-Gordon discrete equations. In this

connection see also [12], [29] and [30]. In [27], we have shown explicit form of one-field

lattice integrable hierarchies including the Itoh-Narita-Bogoyavlenskii lattice hierarchy.

In [28] we derived a class of lattice integrable hierarchies related by some Miura-type

transformation to the Itoh-Narita-Bogoyavlenskii lattice. We remark that this class of

hierarchies contains in particular Yamilov lattice hierarchy [36].

We believe that multi-variate polynomials T ks and Sk

s given by (7) and (14) and

their l-reduced versions could be widely applicable in the theory of classical integrable

systems. In the following two sections we are going to support this point by constructing

discrete equations having a Lax representation in terms of these polynomials and

showing then the sense of these discrete equations from the point of view of lattice

integrable hierarchies.

3. Discrete equations admitting a Lax pair representation

This section is designed to show infinite and finite systems of discrete equations which

share the property of having Lax pair representation. We define our discrete systems

with the help of some generating equations on formal pseudo-difference operators.


3.1. The first class of discrete equations

Consider a partition Ts = (Ts)+,k + (Ts)−,k, where

(Ts)+,k ≡ Λ−sh +k∑

j=1


and

(Ts)−,k ≡∑

j>k

z−jT js (i− (j − 1)n− sh)Λ−sh−jn.

Observe that

[(Ts)+,k, T1] =∑

j≥1

z−k−jνj(i)Λ−(s+1)h−(k+j)n,

where

νj(i) = T k+j(i− (s+ 1)h− (k + j − 1)n)

+k∑

q=1

T qs (i− sh− (q − 1)n)T k+j−q(i− (s+ 1)h− (k + j − 1)n)

− T k+j(i− h− (k + j − 1)n)

−k∑

q=1

T qs (i− (s+ 1)h− (k + j − 1)n)T k+j−q(i− h− (k + j − q − 1)n).

Let

(Ts)+,k = (Ts)+,k +∑

j≥1

z−jhcj(Ts+jn)+,k−jh.

As can be checked

(Ts)+,k = Λ−sh +

k∑

j=1


and the commutativity equation

[(Ts)+,k, T1] = 0 (30)

generate an infinite number of discrete equations

T k+j(i+ sh) +

k∑

q=1

T qs (i)T

k+j−q(i+ sh+ qn)

= T k+j(i) +

k∑

q=1

T qs (i+ h+ (k + j − q)n)T k+j−q(i), j ≥ 1. (31)

Clearly, this system admits an l-reduction with the help of condition T j ≡ 0 for j ≥ l+1

yielding therefore an l-field system of ordinary difference equations.


Commutativity equation (30) appears as a condition of the compatibility of two

linear equations

(Ts)+,k(ψi) =w

zkψi, T1(ψi) = ψi

the first of which we can rewrite as

wψi+kn+sh = zkψi+kn +k∑

j=1

zk−jT js (i+ (k − j + 1)n)ψi+(k−j)n. (32)

Remark that w here is the second spectral parameter. The equation T1(ψi) = ψi with

l-reduced operator T1 looks as

zlψi+ln+h = zlψi+ln +l∑

j=1

zl−jT j(i+ (l − j + 1)n)ψi+(l−j)n. (33)

As a consequence of (30) we have

[(Ts)+,k,S1] = 0

which gives an infinite number of discrete equations

Sk+j(i+ sh) +

k∑

q=1

(−1)qT qs (i+ h)Sk+j−q(i+ sh+ qn)

= Sk+j(i) +

k∑

q=1

(−1)qT qs (i+ (k + j − q)n)Sk+j−q(i), j ≥ 1. (34)

This system also admits an l-reduction with the help of condition Sj ≡ 0 for j ≥ l + 1.

Linear problem S1(ψi) = ψi, in this case, becomes

zlψi+ln−h = zlψi+ln +

l∑

j=1

(−1)jzl−jSj(i+ (l − j + 1)n− h)ψi+(l−j)n. (35)

3.2. The second class of discrete equations

Consider a partition Ss = (Ss)+,k + (Ss)−,k, where

(Ss)+,k ≡ Λsh +

k∑

j=1

(−1)jz−jSjs(i− (j − 1)n)Λsh−jn

and

(Ss)−,k ≡∑

j>k

(−1)jz−jSjs(i− (j − 1)n)Λssh−jn.

Observe that

[(Ss)+,k, T1] =∑

j≥1

z−k−jµj(i)Λ(s−1)h−(k+j)n, (36)


with

µj(i) = T k+j(i+ (s− 1)h− (k + j − 1)n)

+

k∑

q=1

(−1)qSqs (i− (q − 1)n)T k+j−q(i+ (s− 1)h− (k + j − 1)n)

− T k+j(i− h− (k + j − 1)n)

−k∑

q=1

(−1)qSqs (i− h− (k + j − 1)n)T k+j−q(i− h− (k + j − q − 1)n).

Let

(Ss)+,k ≡ (Ss)+,k +∑

j≥1

z−jhHj(Ss−jn)+,k−jh

then

(Ss)+,k = Λsh +k∑

j=1

(−1)jz−jSjs(i− (j − 1)n)Λsh−jn.

We define the second class of discrete equations with the help of commutativity equation

[(Ss)+,k, T1] = 0, (37)

which yields an infinite number of discrete equations

T k+j(i+ sh) +

k∑

q=1

(−1)qSqs (i+ h + (k + j − q)n)T k+j−q(i+ sh)

= T k+j(i) +

k∑

q=1

(−1)qSqs (i)T

k+j−q(i+ qn), j ≥ 1. (38)

As a consequence of (37) we have Lax equation

[(Ss)+,k,S1] = 0, (39)

which gives

Sk+j(i+ sh) +

k∑

q=1

Sqs(i+ (k + j − q)n)Sk+j−q(i+ sh)

= Sk+j(i) +k∑

q=1

Sqs(i+ h)Sk+j−q(i+ qn), j ≥ 1. (40)

Clearly, commutativity equations (37) and (39) appear as a conditions of the

consistency of the linear equation

(Ss)+,k(ψi) =w

zkψi

which we can rewrite as

wψi−sh+kn = zkψi+kn+

k∑

j=1

(−1)jzk−jSjs(i−sh+(k− j+1)n)ψi+(k−j)n(41)

with T1(ψi) = ψi and S1(ψi) = ψi, respectively.


3.3. Remarks on Lax pairs

We have constructed infinite families of ordinary difference systems (31), (34), (38) and

(40), which are equivalent to the consistency conditions of the corresponding pairs of

linear equations. We suppose that Sks and T k

s are expressed via the set of the fields

{T j} in equations (31) and (38), while in (34) and (40) they are expressed via {Sj}.

Let us remember that these two sets of dynamical variables are related to each other

by involutive invertible transformation. In particular, we are interested in l-reduced

versions of these systems which are defined by condition T j ≡ 0, j ≥ l + 1 for (31)

and (38) and Sj ≡ 0, j ≥ l + 1 for (34) and (40). The first question which naturally

arise is: how do we construct the first integrals of the system under consideration using

corresponding pair of linear equations?

Consider a pair of the linear equations

ψi+N1 = L1(ψi, . . . , ψi+N1−1)

and

ψi+N2 = L2(ψi, . . . , ψi+N2−1),

for which the compatibility condition is equivalent to some system of ordinary difference

equations. For example, in the case (32) and (33), we have N1 = kn+sh andN2 = ln+h.

Choosing any of these equations, say the first one, we construct the vector-function

Ψ(i) = (ψ(0)(i), . . . , ψ(N1−1)(i))T , where ψ(j)(i) ≡ ψi+j . Then we use the rest equation

to construct linear equation L(i)Ψ(i) = 0 with some matrix-function L. The second

equation of the form Ψ(i+ 1) = A(i)Ψ(i) is easily defined by

ψ(j)(i+ 1) = ψ(j+1)(i), j = 1, . . . , N1 − 2

and

ψ(N1−1)(i+ 1) = L1(ψ(0)(i), . . . , ψ(N1−1)(i)).

Therefore the system under consideration are equivalent to the Lax equation

L(i+ 1)A(i) = A(i)L(i).

The condition detL = 0 gives us the spectral curve P (w, z) = 0.

3.4. Integrable hierarchies of lattice evolution equations

From (36) we see that

[(Ssn)+,sh, T1] =∑

j≥1

z−sh−jµj(i)Λ−h−jn.

This means that we can correctly define evolution equations by generating relation

∂sT1 = zsh[(Ssn)+,sh, T1] (42)

with

(Ssn)+,sh = Λshn +

sh∑

j=1

(−1)jz−jSjsn(i− (j − 1)n)Λshn−jn.


This generating equation gives an infinite number of evolution differential-difference

equations

∂sTj(i) = T sh+j(i) +

sh∑

q=1

(−1)qSqsn(i+ h+ (j − q)n)T sh+j−q(i)

− T sh+j(i− shn)−sh∑

q=1

(−1)qSqsn(i− shn)T sh+j−q(i− (sh− q)n). (43)

As a consequence of Lax equation (42) we have

∂sS1 = zsh[(Ssn)+,sh,S1]

which yields

∂sSj(i) = (−1)sh

{

Ssh+j(i) +sh∑

q=1

Sqsn(i+ (j − q)n)Ssh+j−q(i)

}

− (−1)sh

{

Ssh+j(i− shn)

+

sh∑

q=1

Sqsn(i− shn + h)Ssh+j−q(i− (sh− q)n)

}

. (44)

Observe that{

∂s +

s−1∑

q=1

Hq∂s−q

}

T1 = zsh[(Ssn)+,sh, T1].

From the latter we see that the stationarity equation{

∂s +

s−1∑

q=1

Hq∂s−q

}

T1 = 0

is equivalent to a special case of (37).

3.5. Some examples of lattice integrable hierarchies

Using (43) and (44), we are able to present the explicit form for a number of lattice

integrable hierarchies.

3.5.1. Itoh-Narita-Bogoyavlenskii lattice hierarchy Let h = 1 while n ≥ 1 is an

arbitrary integer. Lax equation (42) with the operator T1 = Λ−1 + z−1T (i − 1)Λ−1−n,

where T (i) ≡ T 1(i), gives evolution equations

∂sT (i) = (−1)sT (i) {Sssn(i− (s− 1)n+ 1)− Ss

sn(i− sn)} (45)

constituting an integrable hierarchy for the extended Volterra equation [21]

∂1T (i) = T (i)

{

n∑

j=1

T (i− j)−n∑

j=1

T (i+ j)

}

also known as the Itoh-Narita-Bogoyavlenskii lattice [16], [21], [6].


3.5.2. Bogoyavlenskii lattice hierarchy Now let n = 1 while h ≥ 1 is an arbitrary

integer. Lax equation (42) with the operator T1 = Λ−h + z−1Ti−hΛ−h−1 generates

integrable hierarchy

∂sT (i) = (−1)shT (i){

Sshs (i− (s− 1)h+ 1)− Ssh

s (i− sh)}

for the Bogoyavlenskii lattice

∂1T (i) = (−1)hT (i)

{

h∏

j=1

T (i+ j)−h∏

j=1

T (i− j)

}

.

3.5.3. Lattice Gel’fand-Dickii or (l + 1)-KdV hierarchy Let h = 1 and n = 1, while

l ≥ 1 be an arbitrary integer. With the operator

T1 = Λ−1 +

l∑

j=1

z−jT j(i− j)Λ−1−j

we obtain the evolution equations

∂sTj(i) = (−1)s+j

{

l∑

q=j

(−1)qSs+j−qs (i+ q − s+ 1)T q(i)

}

− (−1)s+j

{

l∑

q=j

(−1)qSs+j−qs (i− s)T q(i+ j − q)

}

for the lattice (l+1)-KdV hierarchy [1], [3], [4]. In the case l = 1, we obtain the Volterra

lattice hierarchy

∂sT (i) = (−1)sT (i) {Sss(i− s+ 2)− Ss

s(i− s)}

which, as is known, gives one of the lattice analogues of the KdV hierarchy. In the case

l = 2, we obtain the lattice Boussinesq hierarchy in the form

∂sT1(i) = (−1)s

{

Sss(i− s+ 2)T 1(i)− Ss−1

s (i− s+ 3)T 2(i)}

− (−1)s{

Sss(i− s)T 1(i)− Ss−1

s (i− s)T 2(i− 1)}

,

∂sT2(i) = (−1)sT 2(i) {Ss

s(i− s+ 3)− Sss(i− s)} .

whose the first flow is given by [3]

∂1T1(i) = T 2(i)− T 2(i− 1) + T 1(i)

{

T 1(i− 1)− T 1(i+ 1)}

,

∂1T2(i) = T 2(i)

{

T 1(i− 1)− T 1(i+ 2)}

.


We have not discussed above a quite important question concerning the Hamiltonian,

and what is more important from different points of view, the bi-Hamiltonian structure

of lattice integrable hierarchies with a finite and infinite number of fields given by (43)

and (44). In the previous subsection we have only presented some examples of lattice

integrable hierarchies known in the literature in their explicit form. One can find studies

on the Hamiltonian approach to lattice systems, for example, in [1], [3] - [5], [7], [9] and

[35].


4. Integrable hierarchies of lattice evolution equations from the point of

view of the KP hierarchy

In the previous section we have shown some classes of discrete equations with their Lax

representation in terms of polynomials T ks and Sk

s which were given in explicit form by

(7), (14), (18) and (19). In this section we have a look at that from another point of view.

In [23]-[27] we developed an approach which relates some evolutionary lattice equations

and their hierarchies to the KP hierarchy. We state in this section this approach. This

aims, in particular, to clarify the appearance of l-reduced versions of discrete equations

(31), (34), (38) and (40).

4.1. Free sequences of KP hierarchies. Invariant submanifolds

To begin with, we consider the free sequences of KP hierarchies {h(i, z) : i ∈ Z} in the

form of local conservation laws [11], [34], that is,

∂sh(i, z) = ∂H(s)(i, z), ∀i ∈ Z,

where the formal Laurent series h(z) = z+∑

j≥2 hjz−j+1 and H(s)(z) = zs+

∑

j≥1Hsj z

−j

are related to the KP wave function by h(z) = ∂ψ/ψ and H(s)(z) = ∂sψ/ψ, respectively.

Therefore, each coefficient Hsj is supposed to be some differential polynomial Hs

j =

Hsj [h2, . . . , hs+k]. More exactly, the Laurent series H(s)(z) is calculated as a projection

of zs on the linear space H+ =⟨

1, h, h(2), ...⟩

spanned by Faa di Bruno iterates

h(k) ≡ (∂ + h)k(1) . For example,

H(1) = h, H(2) = h(2) − 2h2, H(3) = h(3) − 3h2h− 3h3 − 3h′2

etc. Remark, that h(s)(z) = ∂sψ/ψ. As is known, dynamics of the coefficients of Hsj in

virtue of KP flows is defined by the invariance property(

∂s +H(s))

H+ ⊂ H+, ∀s ≥ 1. (46)

Conversely, it can be seen, that a hierarchy of dynamical systems governed by (46),

known as the Central System, is equivalent to the KP hierarchy [10]. Moreover, we

consider also a Laurent series a(i, z) = z+∑

j≥1 aj(i)z−j+1 ≡ zψi+1/ψi. By its definition,

it satisfies the following evolution equations

∂sa(i, z) = a(i, z){

H(s)(i+ 1, z)−H(s)(i, z)}

, (47)

which in turn can be presented in the form of differential-difference conservation laws

∂sξ(i, z) = H(s)(i+ 1, z)−H(s)(i, z)

with

ξ(i, z) = ln a(i) = ln z −∑

j≥1

1

j

(

−∑

k≥1

ak(i)z−k

)j

≡ ln z +∑

j≥1

ξj(i)z−j .

It is convenient to consider a free bi-infinite sequence of KP hierarchies as two equations

[19]

∂shk(i) = ∂Hsk−1(i), ∂sξk(i) = Hs

k(i+ 1)−Hsk(i). (48)


The two theorems given below yield an infinite number of invariant submanifolds

of the phase space of the sequence of KP hierarchies whose points are defined by

{ak(i), Hsk(i) : i ∈ Z}.

Theorem 1 [23] Submanifold Snk−1 defined by the condition

zk−na[n](i, z) ∈ H+(i), ∀i ∈ Z (49)

is invariant with respect to the flows given by evolution equations (48).

Here, by definition,

a[s](i, z) =

∏s

j=1 a(i+ j − 1, z), for s ≥ 1;

1, for s = 0;∏|s|

j=1 a−1(i− j, z), for s ≤ −1.

The condition (49) can be ultimately rewritten as a single generating relation∗

zk−na[n] = H(k) +

k∑

j=1

a[n]j H

(k−j), (50)

where a[s]j are defined as the coefficients of Laurent series a[s] = zs +

∑

j≥1 a[s]j z

s−j being

ultimately some quasi-homogeneous polynomials a[s]j = a

[s]j [a1, . . . , aj].

Theorem 2 [24] The following chain of inclusions

Sns−1 ⊂ S2n

2s−1 ⊂ S3n3s−1 ⊂ · · · (51)

is valid.

4.2. The nth discrete KP hierarchy

Let us consider the restriction of (48) on Sn0 which is defined by the condition

z1−na[n] = h+a[n]1 . Clearly, by this constraint we should get as a result some differential-

difference equations on infinite number of dynamical variables ak = ak(i) in the form

of differential-difference conservation laws, but there is a need to know how coefficients

Hsj are expressed via aj for all s ≥ 1. By our assumption, we already know that

H1k ≡ hk+1 = a

[n]k+1. To obtain explicit expressions for all Hs

k we must use theorem 2.

For s = 1, we have a chain

Sn0 ⊂ S2n

1 ⊂ S3n2 ⊂ · · ·

Solving, step-by-step, the defining equations for all invariant submanifolds in this chain

we come to [24]

Hsk = F

(n,s)k [a1, . . . , ak+s] ≡ a

[sn]k+s +

s−1∑

j=1

q(n,sn)j a

[(s−j)n]k+s−j . (52)

∗ Note that we sometimes do not indicate dependence on discrete variable i in formulas which contain

no shifts with respect to this variable.


Here q(n,r)j , being some quasi-homogeneous polynomials, are defined through the relation

zr = a[r] +∑

j≥1

q(n,r)j zj(n−1)a[r−jn]. (53)

We can code all relations (52) in one generating relation

H(s) = F (n,s) = zs−sn

{

a[sn] +

s∑

j=1

zj(n−1)q(n,sn)j a[(s−j)n]

}

, (54)

where F (n,s) ≡ zs +∑

j≥1 F(n,s)j z−j . Thus the restriction of dynamics given by (48) on

Sn0 yields evolution equations in the form of differential-difference conservation laws

∂sξk(i) = F(n,s)k (i+ 1)− F

(n,s)k (i) (55)

on an infinite number of fields {ak = ak(i)}. Let M be the corresponding phase space.

We call the hierarchy of the flows on M given by evolution equations (55) the nth

discrete KP hierarchy. Let us observe that in terms of KP wave function the relation

(54) takes the form

∂sψi = zsψi+sn +s∑

j=1

zs−jq(n,sn)j (i)ψi+(s−j)n, (56)

while (53) is equivalent to

zrψi = zrψi+r +∑

j≥1

zr−jq(n,r)j (i)ψi+r−jn.

As a condition of consistency of these linear equations we obtain the evolution equations

∂sq(n,r)k (i) = q

(n,r)k+s (i+ sn) +

s∑

j=1

q(n,sn)j (i)q

(n,r)k+s−j(i+ (s− j)n)−

− q(n,r)k+s (i)−

s∑

j=1

q(n,sn)j (i+ r − (k + s− j)n)q

(n,r)k+s−j(i) (57)

which, by their construction, are identities by virtue of the nth discrete KP hierarchy

(55).

4.3. Some identities for q(n,s)j and a

[s]j

Relation (53) yields the infinite triangle system

a[r]k +

k−1∑

j=1

a[r−jn]k−j q

(n,r)j + q

(n,r)k = 0, k ≥ 1 (58)

from which we can calculate, for example,

q(n,r)1 = −a[r]1 , q

(n,r)2 = −a[r]2 + a

[r]1 a

[r−n]1 ,

q(n,r)3 = −a[r]3 + a

[r]1 a

[r−n]2 + a

[r−2n]1 a

[r]2 − a

[r]1 a

[r−n]1 a

[r−2n]1


etc. It was shown in [27] that if (58) is valid then a more general identity

a[p]k (i) = a

[r]k (i) +

k−1∑

j=1

a[r−jn]k−j (i)q

(n,r−p)j (i+ p) + q

(n,r−p)k (i+ p), (59)

for all integers r and p, holds. Resolving the latter in favor of q(n,r−p)k (i+ p) yields

q(n,r−m)k (i+m) = a

[m]k (i) +

k−1∑

j=1

q(n,r−(k−j)n)j (i)a

[m]k−j(i) + q

(n,r)k (i).

We remark that polynomials a[s]j , by definition, must satisfy

a[s1+s2]k (i) = a

[s1]k (i) +

k−1∑

j=1

a[s1]j (i)a

[s2]k−j(i+ s1) + a

[s2]k (i+ s1)

= a[s2]k (i) +

k−1∑

j=1

a[s2]j (i)a

[s1]k−j(i+ s2) + a

[s1]k (i+ s2) (60)

while for q(n,s)j we have

q(n,s1+s2)k (i) = q

(n,s1)k (i) +

k−1∑

j=1

q(n,s1)j (i)q

(n,s2)k−j (i+ s1 − jn) + q

(n,s2)k (i+ s1)

= q(n,s2)k (i) +

k−1∑

j=1

q(n,s2)j (i)q

(n,s1)k−j (i+ s2 − jn) + q

(n,s1)k (i+ s2).

4.4. Restriction of the nth discrete KP hierarchy on submanifold Mn,p,l

To obtain in this framework the description of some integrable differential-difference

systems with a finite number of fields, we consider the intersection Sn0 ∩Sp

l−1. Obviously,

it is equivalent to the restriction of the flows of the nth discrete KP hierarchy on a

submanifold Mn,p,l ⊂ M. Taking into account (50) and (52), we immediately obtain

that defining equations for Mn,p,l are coded in the relation

zl−pa[p] = F (n,l) +l∑

j=1

a[p]j F

(n,l−j)

from which we derive

J(n,p,l)k [a1, . . . , ak+l] = 0, ∀k ≥ 1 (61)

with

J(n,p,l)k = a

[p]k+l − F

(n,l)k −

l−1∑

j=1

a[p]j F

(n,l−j)k

= a[p]k+l(i)− a

[ln]k+l(i)−

l−1∑

j=1

q(n,ln−p)j (i+ p)a

[(l−j)n]k+l−j (i). (62)


We see from (62) that J(n,ln,l)k = 0 is just an identity and, therefore, produces no invariant

submanifold. This is because Sn0 ⊂ S ln

l−1 by virtue of theorem 2.

Let us observe that, taking into account (59), we can rewrite (62) as

J(n,p,l)k = Q

(n,p,l)k +

k−1∑

j=1

a[−(k−j)n]j Q

(n,p,l)k−j , (63)

where Q(n,p,l)k (i) ≡ q

(n,ln−p)k+l (i+ p). Resolving this quasi-linear infinite triangle system in

favor of Q(n,p,l)k gives

Q(n,p,l)k = J

(n,p,l)k +

k−1∑

j=1

q(n,−(k−j)n)j J

(n,p,l)k−j (64)

for all k ≥ 1. This means that, in principle, we can define the submanifold Mn,p,l by

equations Q(n,p,l)k [a1, . . . , ak+l] = 0.

4.5. Linear equations on KP wave function corresponding to submanifold Mn,p,l

Some remarks about linear equations on the KP formal wave function ψi which follows

as a result of the restriction of the nth discrete KP hierarchy on the submanifold Mn,p,l

are in order. Let J (n,p,l)(z) ≡∑

j≥1 J(n,p,l)j z−j . We observe that an infinite number of

equations (61) can be presented by single a generating relation

J (n,p,l)(i, z) = zl−pa[p](i, z)

−zl−ln

{

a[ln](i, z) +l∑

j=1

zj(n−1)q(n,ln−p)j (i+ p)a[(l−j)n](i, z)

}

= 0.

Clearly, in terms of KP wave function, we can rewrite the latter relation as

J (n,p,l)(i, z) =zlψi+p − zlψi+ln −

∑l

j=1 zl−jq

(n,ln−p)j (i+ p)ψi+(l−j)n

ψi

= 0.

Thus in terms of the KP wave function the restriction of nth discrete KP hierarchy on

Mn,p,l is given by the linear equation

zlψi+ln +

l∑

j=1

zl−jq(n,ln−p)j (i+ p)ψi+(l−j)n = zlψi+p. (65)

Second linear equation which we should have in mind is (56).

What we obtain is that on Mn,p,l the KP wave function satisfies a linear equation

T1(ψi) = ψi with the l-reduced operator T1 in the case ln− p ≤ −1 while if ln− p ≥ 1

then it satisfies the equation S1(ψi) = ψi with l-reduced operator S1. So, let

h =

{

p− ln if ln− p ≤ −1;

ln− p if ln− p ≥ 1.

Let us denote

q(n,ln−p)j (i) = T j(i− h− (j − 1)n), j = 1, . . . , l


in the first case and

q(n,ln−p)j (i) = (−1)jSj(i− (j − 1)n), j = 1, . . . , l

in the second case, respectively. Substituting the latter into (65) gives (33) and (35).

We observe that in both cases

q(n,−sh)j (i) = T j

s (i− sh− (j − 1)n) (66)

and

q(n,sh)j (i) = (−1)jSj

s(i− (j − 1)n). (67)

Let us turn now to equation (56) defining an infinite number of flows of the nth

discrete KP hierarchy and consider only the flows ∂sh. With (67) we obtain

∂shψi = zshψi+shn +

sh∑

j=1

zsh−jq(n,shn)j (i)ψi+(sh−j)n

= zsh

{

ψi+shn +

sh∑

j=1

(−1)jz−jSjsn(i− (j − 1))ψi+(sh−j)n

}

= zsh (Ssn)+,sh (ψi). (68)

Let us summarize what we have said above. Restricting the nth discrete KP

hierarchy to Mn,p,l or more exactly its sub-hierarchy {∂sh : s ≥ 1}, where h is defined by

the triple (n, p, l) as above, leads to l-reduced lattice systems (43) or (44) in dependence

of the sign of ln − p. In the following two subsections we show how discrete equations

(31), (34), (38) and (40) appear in this framework and therefore we clarify their meaning.

4.6. Weaker invariance conditions

We have shown above that some integrable lattices can be obtained by imposig some

suitable conditions compatible with the flows of the nth discrete KP hierarchy on Mn,p,l

and presented some examples known in the literature. In this section, we are going to

show that there exist weaker invariance conditions. To this aim, we observe that the

relations

∂sQ(n,p,l)k (i) = Q

(n,p,l)s+k (i+ sn) +

s∑

j=1

q(n,sn)j (i+ p)Q

(n,p,l)s+k−j(i+ (s− j)n)

−Q(n,p,l)s+k (i)−

s∑

j=1

q(n,sn)j (i− (s+ k − j)n)Q

(n,p,l)s+k−j(i) (69)

are valid by virtue of the nth discrete KP hierarchy. Let {I(n,p,l)k : k ≥ 1} be some infinite

set of quasi-homogeneous polynomials related with {Q(n,p,l)k : k ≥ 1} by relations

Q(n,p,l)k = I

(n,p,l)k +

k−1∑

j=1

ξk−1,jI(n,p,l)k−j (70)


with some unknown coefficients ξk,s to be defined . Consider equation (69) on Q(n,p,l)1 =

I(n,p,l)1 , that is,

∂sQ(n,p,l)1 (i) = Q

(n,p,l)s+1 (i+ sn) +

s∑

j=1

q(n,sn)j (i+ p)Q

(n,p,l)s+1−j(i+ (s− j)n)

−Q(n,p,l)s+1 (i)−

s∑

j=1

q(n,sn)j (i− (s+ 1− j)n)Q

(n,p,l)s+1−j(i). (71)

Substituting (70) into the right-hand side of (71) gives

∂sI(n,p,l)1 (i) = I

(n,p,l)s+1 (i+ sn) +

s∑

j=1

ξs,j(i+ sn)I(n,p,l)s−j+1(i+ sn) + q

(n,sn)1 (i+ p)

×

{

I(n,p,l)s (i+ (s− 1)n) +s−1∑

j=1

ξs−1,j(i+ (s− 1)n)I(n,p,l)s−j (i+ (s− 1)n)

}

+ · · ·

+ q(n,sn)s (i+ p)I(n,p,l)1 (i)− I

(n,p,l)s+1 (i)−

s∑

j=1

ξs,j(i)I(n,p,l)s−j+1(i)− q

(n,sn)1 (i− sn)

×

{

I(n,p,l)s (i) +

s−1∑

j=1

ξs−1,j(i)I(n,p,l)s−j (i)

}

− · · · − q(n,sn)s (i− n)I(n,p,l)1 (i).

Impose now the periodicity condition Ik(i+n) = Ik(i) for all k ≥ 1. With this condition

∂sI(n,p,l)1 (i) =

{

ξs,1(i+ sn) + q(n,sn)1 (i+ p)− ξs,1(i)− q

(n,sn)1 (i− sn)

}

I(n,p,l)s (i)

+{

ξs,2(i+ sn) + q(n,sn)1 (i+ p)ξs−1,1(i+ (s− 1)n) + q

(n,sn)2 (i+ p)

−ξs,2(i)− q(n,sn)1 (i− sn)ξs−1,1(i)− q

(n,sn)2 (i− (s− 1)n)

}

I(n,p,l)s−1 (i) + · · ·

+

{

ξs,s(i+ sn) +

s−1∑

j=1

q(n,sn)j (i+ p)ξs−j,s−j(i+ (s− j)n) + q(n,sn)s (i+ p)

−ξs,s(i)−s−1∑

j=1

q(n,sn)j (i− (s− j + 1)n)ξs−j,s−j(i)− q(n,sn)s (i− n)

}

I(n,p,l)1 (i).

Now we observe that if ξs,k(i) = q(n,−p−(s−k+1)n)k (i+ p), then the coefficients at I

(n,p,l)j in

the last relation become identically zeros. So, let

Q(n,p,l)k (i) = I

(n,p,l)k (i) +

k−1∑

j=1

q(n,−p−(k−j)n)j (i+ p)I

(n,p,l)k−j (i). (72)

One can check that resolving these relations in favor of I(n,p,l)k yields

I(n,p,l)k (i) = Q

(n,p,l)k (i) +

k−1∑

j=1

a[−p−(k−j)n]j (i+ p)Q

(n,p,l)k−j (i). (73)

Taking into account (64), we derive from (73)

I(n,p,l)k (i) = J

(n,p,l)k (i) +

k−1∑

j=1

a[−p]j (i+ p)J

(n,p,l)k−j (i) (74)


and

J(n,p,l)k = I

(n,p,l)k +

k−1∑

j=1

a[p]j (i)I

(n,p,l)k−j .

In terms of the Laurent series relations, (74) takes the form

I(n,p,l)(i, z) = zpa[−p](i+ p, z)J (n,p,l)(i, z)

=ψi

ψi+p

J (n,p,l)(i, z)

=zlψi+p − zlψi+ln −

∑l

j=1 zl−jq

(n,ln−p)j (i+ p)ψi+(l−j)n

ψi+p

.

Therefore, we have proved the following. If all I(n,p,l)k defined by (73) satisfy the

periodicity condition

I(n,p,l)k (i+ n) = I

(n,p,l)k (i) (75)

then I(n,p,l)1 do not depend of all evolution parameters ts. Using similar reasoning,

step-by-step, we can show that under this condition in fact all I(n,p,l)k for k ≥ 1 are

constants. It is evident that by virtue of (72), the equations I(n,p,l)k ≡ 0 define the

invariant submanifold Mn,p,l while the periodicity condition (75) is weaker than the

latter condition. We observe that the coefficients of equations (69) considered as quasi-

linear equations with respect to Q(n,p,l)k do not depend on l. This means that periodicity

conditions I(n,p,l)k (i+ n) = I

(n,p,l)k (i) where

I(n,p,l)k ≡ I

(n,p,l)k +

l∑

j=1

hjI(n,p,l−j)k (76)

with arbitrary constants hj also select some invariant submanifold. Let d be some divisor

of the number n, then, it is obvious that the periodicity conditions

I(n,p,l)k (i+ d) = I

(n,p,l)k (i) (77)

define some invariant submanifold N dn,p,l of the nth discrete KP hierarchy. Let us

formulate all the above-mentioned in the following theorem.

Theorem 3 Infinite number of the periodicity conditions (77), where polynomials I(n,p,l)k

are defined by (73) and (76) and d is some divisor of n, are compatible with the flows

of the nth discrete KP hierarchy.

4.7. Discrete equations (31), (34), (38) and (40) as reductions of evolution

differential-difference equations

Here, we use the quite general theorem 3 to construct classes of constraints compatible

with integrable hierarchies corresponding to different submanifolds Mn,p,l . Let the


triple of integers (n, p, k) be such that kn − p = sh for some arbitrary s ≥ 0, then by

virtue of (67)

I(n,p,k)(i, z) =zkψi+p − zkψi+kn −

∑k

j=1 zk−jq

(n,kn−p)j (i+ p)ψi+(k−j)n

ψi+p

=zkψi+p − zkψi+kn −

∑k

j=1(−1)jzk−jSjs(i+ p− (j − 1)n)ψi+(k−j)n

ψi+p

.

Replacing k 7→ k − qh, s 7→ s− qn and therefore p 7→ p, we obtain

I(n,p,k−qh)(i, z) =

{

zk−qhψi+p − zk−qhψi+(k−qh)n

−

k−qh∑

j=1

(−1)jzk−qh−jSjs−qn(i+ p− (j − 1)n)ψi+(k−qh−j)n

}

/ψi+p

and consequently

I(n,p,k)(i, z) ≡ I(n,p,k)(i, z) +κ∑

q=1

HqI(n,p,k−qh)(i, z)

=

{(

zk +κ∑

q=1

Hqzk−qh

)

ψi+p

−zkψi+kn −k∑

j=1

(−1)jzk−jSjs(i+ p− (j − 1)n)ψi+(k−j)n

}

/ψi+p. (78)

Impose the condition

I(n,p,k)(i, z) =∑

j≥1

Ijz−j ,

where Ij’s are some constants. According to theorem 3 this condition is invariant with

respect to the flows of the nth discrete KP hierarchy. One sees that (78) becomes linear

problem (41), with

w = zk +

κ∑

q=1

Hqzk−qh −

∑

j≥1

Ijz−j .

By analogy, let the triple of integers (n, p, k) be such that kn− p = −sh for some

arbitrary s ≥ 0. Then by virtue of (66)

I(n,p,k)(i, z) =zkψi+p − zkψi+kn −

∑k

j=1 zk−jq

(n,kn−p)j (i+ p)ψi+(k−j)n

ψi+p

=zkψi+p − zkψi+kn −

∑k

j=1 zk−jT j

s (i+ (k − j + 1)n)ψi+(k−j)n

ψi+p

.

Imposing invariant condition

I(n,p,k)(i, z) ≡ I(n,p,k)(i, z) +κ∑

q=1

cqI(n,p,k−qh)(i, z) =

∑

j≥1

Ijz−j

leads to linear problem (32).


4.8. Examples

4.8.1. One-field difference systems In the case Mn,n+1,1 we have ln − p = −1 and

therefore h = p−ln = 1. In this situation the nth discrete KP hierarchy is reduced to the

Itoh-Narita-Bogoyavlenskii lattice given by (45). Two classes of compatible constraints

are deduced as the one-field reduction of discrete equations (31) and (38), that is,

T ks (i)T (i+ s+ kn) = T k

s (i+ n+ 1)T (i)

and

Sks (i)T (i+ kn) = Sk

s (i+ n+ 1)T (i+ s),

respectively, with

T ks (i) = T k

s (i) +

k∑

j=1

cjTk−js+jn(i)

and

Sks (i) = Sk

s (i) +k∑

j=1

(−1)jHjSk−js−jn(i+ jn).

Remember that we suppose here that s − jn ≤ 1. A more general class of one-field

lattices is given by

T ks (i)T (i+ sh + kn) = T k

s (i+ n+ h)T (i) (79)

and

Sks (i)T (i+ kn) = Sk

s (i+ n+ h)T (i+ sh), (80)

where h and n are supposed to be co-prime positive integers. Both the equations

(80) and (79) are of the form of the autonomous difference equation (1) with the

order N = sh + kn whose right-hand side is a rational function of its arguments

and corresponding parameters. Let us observe that making use of the identities for

polynomials T ks and Sk

s we can rewrite discrete equations (80) and (79) in the following

equivalent form:

T ks−1(i+ h)T (i+ sh + kn) = T k

s−1(i+ n+ h)T (i) (81)

and

Sks+1(i)T (i+ kn) = Sk

s+1(i+ n)T (i+ sh). (82)

It is evident from this that in the case n = 0 and h = 1 which corresponds to symmetric

multi-variate polynomials T ks {yj} and Sk

s {yj}, equations (82) and (81) are specified only

as the periodicity equation T (i+ s) = T (i).


4.8.2. The case h = n = 1 The simplest case, when we can obtain nontrivial discrete

nonlinear equations different from the periodicity one is h = n = 1. In this case (82)

and (81) looks as

T ks−1(i+ 1)T (i+ s+ k) = T k

s−1(i+ 2)T (i) (83)

and

Sks+1(i)T (i+ k) = Sk

s+1(i+ 1)T (i+ s). (84)

What we know is that these equations play the role of compatible constraints for

the Volterra lattice hierarchy defining invariant submanifolds in solution space of the

Volterra lattice. We claim but do not prove here that for an arbitrary fixed pair (k, s)

such that s ≥ k + 1, (84) and (83) represents in fact the same dynamical system. The

point is that one can transform (84) into (83) and back by interchanging {c1, . . . , ck}

and {H1, . . . , Hk} (cf. [22]). We will prove this fact in a subsequent publication.

For spectral curves, our calculations using Lax operators defined by (32) and (33),

in the case h = n = l = 1, yield the following. If s+ k = 2g + 1, where g ≥ k we obtain

H0w2 + zg+1Rg(z)w − z2g+1Rk(z) = 0, (85)

while in the case s+ k = 2g + 2 , where g ≥ k, we get

H0w2 − zg+1Rg+1(z)w + z2g+2Rk(z) = 0, (86)

with Rg(z) = zg +∑g

j=1Hjzg−j and Rk(z) = zk +

∑k

j=1 Hjzk−j, where Hj =

Hj +∑j−1

q=1 cqHj−q + cj. We remark that Hj’s are the first integrals integrals of equation

(83), while cj’s are the parameters entered in these equations. In particular,

H0 =

∏s+k

j=1 Ti+j−1

T ks−1(i+ 1)

. (87)

In both cases we have an expansion

w = zk +k∑

j=1

cjzk−j −

∑

j≥1

Ijz−j ,

where the coefficients Ij are some polynomials of variables Hj and cj.

Let us remark that in both cases we have a hyperelliptic spectral curve of genius g.

For (85), with the help of birational transformation

z =1

Z, w =

W − rg(Z)

2H0Z2g+1, W =

2H0w + zg+1Rg(z)

2z2g+1,

where

rg(Z) ≡ ZgRg

(

1

Z

)

, rk(Z) ≡ ZkRk

(

1

Z

)

,

we get

W 2 = P2g+1(Z) ≡ r2g(Z) + 4H0Z2g−k+1rk(Z),


while for (86), we use

z = Z, w = Zg+1W +Rg+1(Z)

2H0, W =

2H0w − zg+1Rg+1(z)

zg+1

to obtain

W 2 = P2g+2(Z) ≡ R2g+1(Z)− 4H0Rk(Z).

Making use of the transformation

w 7→zg+1

H0

(

zg−kw − Rg(z))

for (85) gives

z2g−2k+1w2 − zg−k+1Rg(z)w −H0Rk(z) = 0 (88)

while the transformation

w 7→zg+1

H0

(

Rg+1(z)− zg−k+1w)

converts (86) to the following form:

z2g−2k+2w2 − zg−k+1Rg+1(z)w +H0Rk(z) = 0. (89)

The reason why we have shown these transformations is that (88) and (89) are nothing

but the spectral curves for Lax operators defined by (33) and (41), in the case

h = n = l = 1. Let us remark that in this case {H1, . . . , Hk} is the set of the parameters

entered in equation (84), while other Hj’s and all cj’s are the first integrals. We will

describe in full detail all these integrals in a subsequent publication.


Let us remark that (84) has obvious integral♯

H0 = (−1)kSks+1(i)

s−1∏

j=k

T (i+ j). (90)

In particular case k = 1 one has

H0 = −{

S1s+1(i)−H1

}

s−1∏

j=1

T (i+ j),

where S1s+1(i) =

∑s

j=0 T (i+ j). The latter equation restricted by the condition H1 = 0

together with its integrals can be found in [12] (eq. (4)). In particular, in that paper

plausible formula was found yielding degrees of mapping iterates suggesting that this

equation has zero algebraic entropy.

♯ Here we use the same symbol H0 as in (87) because as we have already said equations (84) and (83)

for fixed pair (k, s) represent the same dynamical system. Therefore the integral H0 has two different,

at first glance, forms (87) and (90).


5. Discussion

Studies carried out in [12], [17], [27], [28], [29] and in this paper suggest that the

polynomials T ks and Sk

s corresponding to different pairs of co-prime integers n and h

might be quite universal and convenient combinatorial object and therefore we hope

that they deserve attention to investigate them.

Unfortunately, in this paper we have not considered in full detail the important

question of the structure of the first integrals for the obtained ordinary difference

equations and corresponding spectral curves P (w, z) = 0. We hope to fill this gap

in the following publications to investigate one-field equations (80) and (79) and, in

particular, equations (84) and (83). Also, we would like to study the relationship of the

explicit form of integrable hierarchies (43) and (44) with their Hamiltonian structures.

Acknowledgments

I wish to thank the referees for carefully reading the manuscript and for remarks which

enabled the presentation of the paper to be improved. This work was partially supported

by grant NSh-5007.2014.9.

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on some classes of discrete polynomials and ordinary ... · on some classes of discrete polynomials...

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